The Dirichlet to Neumann Map for the modified Helmholtz and Helmholtz Equations with Complex Boundary Data

Abstract

We present a spectral collocation type method for computing the Dirichlet to Neumann map for the modified Helmholtz equation. For regular and irregular polygons, we demonstrate quadratic convergence for sine basis functions and exponential convergence for Chebyshev basis functions.

We go on to outline how our method can be extended to the Helmholtz equation, for which we also present numerical results.

Our work is an extension of previous results of Prof. Fokas and collaborators for the Laplace equation (J. of Comput. and Appl. Maths. 167, 465-483 (2004)).